Mermin–Wagner theorem

Intuitively, this theorem implies that long-range fluctuations can be created with little energy cost, and since they increase the entropy, they are favored.

This preference is because if such a spontaneous symmetry breaking occurred, then the corresponding Goldstone bosons, being massless, would have an infrared divergent correlation function.

The absence of spontaneous symmetry breaking in d ≤ 2 dimensional infinite systems was rigorously proved by David Mermin and Herbert Wagner (1966), citing a more general unpublished proof by Pierre Hohenberg (published later in 1967) in statistical mechanics.

This makes a two dimensional massless scalar field slightly tricky to define mathematically.

If you define the field by a Monte Carlo simulation, it doesn't stay put, it slides to infinitely large values with time.

A random walk also moves arbitrarily far from its starting point, so that a one-dimensional or two-dimensional scalar does not have a well defined average value.

While the Mermin–Wagner theorem prevents any spontaneous symmetry breaking on a global scale, ordering transitions of Kosterlitz–Thouless–type may be allowed.

However, the theorem does not prevent the existence of a phase transition in the sense of a diverging correlation length ξ.

, and a low-temperature phase with quasi-long-range order where G(r) decays according to some power law for "sufficiently large", but finite distance r (a ≪ r ≪ ξ with a the lattice spacing).

We will present an intuitive way[2] to understand the mechanism that prevents symmetry breaking in low dimensions, through an application to the Heisenberg model, that is a system of n-component spins Si of unit length |Si| = 1, located at the sites of a d-dimensional square lattice, with nearest neighbour coupling J.

Consider the low temperature behavior of this system and assume that there exists a spontaneously broken symmetry, that is a phase where all spins point in the same direction, e.g. along the x-axis.

We have whence Ignoring the irrelevant constant term H0 = −JNd and passing to the continuum limit, given that we are interested in the low temperature phase where long-wavelength fluctuations dominate, we get The field fluctuations σα are called spin waves and can be recognized as Goldstone bosons.

To find if this hypothetical phase really exists we have to check if our assumption is self-consistent, that is if the expectation value of the magnetization, calculated in this framework, is finite as assumed.

The average magnetization is and the first order correction can now easily be calculated: The integral above is proportional to and so it is finite for d > 2, but appears to be divergent for d ≤ 2 (logarithmically for d = 2).

In particular [citation needed]: In this general setting, Mermin–Wagner theorem admits the following strong form (stated here in an informal way): When the assumption that the Lie group be compact is dropped, a similar result holds, but with the conclusion that infinite-volume Gibbs states do not exist.

Finally, there are other important applications of these ideas and methods, most notably to the proof that there cannot be non-translation invariant Gibbs states in 2-dimensional systems.

A typical such example would be the absence of crystalline states in a system of hard disks (with possibly additional attractive interactions).

[6] One reason for the lack of global symmetry breaking is, that one can easily excite long wavelength fluctuations which destroy perfect order.

We consider harmonic approximation, where the forces (torque) between neighbouring moments increase linearly with the angle of twisting

If one considers the excited mode with the lowest energy in one dimension (see figure), then the moments on the chain of length

Long range order is not affected by this mode and global symmetry breaking is allowed.

is well defined, the deviations from a perfect periodic chain increase with the square root of the system size.

For the case of two dimensions, Herbert Wagner and David Mermin have proved rigorously, that fluctuations distances increase logarithmically with systems size

These are micrometre-sized particles dispersed in water and sedimented on a flat interface, thus they can perform Brownian motions only within a plane.

The sixfold crystalline order is easy to detect on a local scale, since the logarithmic increase of displacements is rather slow.

A direct experimental proof of Hohenberg–Mermin–Wagner fluctuations would be, if the displacements increase logarithmic with the distance of a locally fitted coordinate frame (blue).

[10][11] This work analyses the recurrence probability of random walks and spontaneous symmetry breaking in various dimensions.

The finite recurrence probability of a random walk in one and two dimension shows a dualism to the lack of perfect long-range order in one and two dimensions, while the vanishing recurrence probability of a random walk in 3D is dual to existence of perfect long-range order and the possibility of symmetry breaking.

Real magnets usually do not have a continuous symmetry, since the spin-orbit coupling of the electrons imposes an anisotropy.

[16] The discrepancy between the Hohenberg–Mermin–Wagner theorem (ruling out long range order in 2D) and the first computer simulations (Alder&Wainwright), which indicated crystallization in 2D, once motivated J. Michael Kosterlitz and David J, Thouless, to work on topological phase transitions in 2D.